Non-Hermitian Walks Reveal Novel Edge State Control and Topology.

The behaviour of quantum particles undergoing a ‘walk’ – a quantum analogue of random walks – presents a fertile ground for exploring topological phenomena, states of matter characterised by robust properties insensitive to local perturbations. Recent research focuses on how these walks evolve in time, specifically when gain and loss are introduced into the system, creating what is known as a non-unitary dynamic. This introduces complexities to the established relationship between a material’s bulk properties and its boundary states, a principle known as bulk-boundary correspondence. Huimin Wang, Zhihao Xu, and Zhijian Li, all from the Institute of Theoretical Physics, State Key Laboratory of Quantum Optics Technologies and Devices, investigate these effects in their article, “Topological Phase Transitions and Edge-State Transfer in Time-Multiplexed Quantum Walks”. Their work details how manipulating the gain and loss within these time-dependent quantum walks allows for a restoration of a generalised bulk-boundary correspondence and reveals a novel mechanism for transferring edge states, potentially offering new avenues for controlling quantum information.

Researchers actively pursue the experimental realisation of complex quantum control schemes, with integrated photonics becoming a favoured platform due to its capacity for precise light manipulation and facilitation of intricate experimental designs. Complementing photonic approaches, cold atom experiments and implementations utilising photonic crystals broaden the range of physical systems investigated, reinforcing the versatility of Floquet engineering. Floquet engineering, a technique employing periodic modulation to control quantum systems, builds upon the principles of driven quantum systems and establishes a foundation for advanced research.

Recent investigations consistently explore the manipulation of quantum systems through periodic modulation, with a particular emphasis on topological and non-Hermitian physics. Non-Hermitian systems, unlike conventional quantum systems described by Hermitian operators, do not conserve probability and exhibit unique properties. These systems display modified topological edge state behaviour and altered spectral signatures, challenging established understandings of quantum mechanics and opening new avenues for materials science and quantum technologies. Theoretical contributions from researchers such as Andersson et al. enhance the understanding of these systems, while Cardano et al. explore the integration of orbital angular momentum with non-Hermitian frameworks, expanding the toolkit for quantum state manipulation. Current review articles consolidate progress in Floquet engineering, while ongoing work continues to refine quantum control and materials science, indicating a dynamic and evolving field.

This research investigates the topological and edge-state properties of a time-multiplexed non-unitary walk, demonstrating a breakdown of the conventional bulk-boundary correspondence in non-unitary regimes and proposing a generalized framework to restore it. The bulk-boundary correspondence, a fundamental principle in topological physics, relates the properties of a material’s interior (bulk) to the behaviour of its surface (boundary), specifically predicting the existence of protected edge states at the boundary.

Researchers constructed a Floquet operator, incorporating tunable gain and loss, and systematically analysed both unitary and non-unitary regimes to reveal the behaviour of edge states under varying conditions. In the unitary case, the conventional bulk-boundary correspondence held, with edge modes localising at opposite boundaries as predicted by topological invariants, confirming expected behaviour in a Hermitian system. However, in the non-unitary regime, non-Hermitian skin effects emerged, leading to a breakdown of the conventional bulk-boundary correspondence, indicating a fundamental shift in system behaviour due to the presence of gain and loss. Non-Bloch band theory and generalized Brillouin zones were then applied to restore a generalized bulk-boundary correspondence, providing a theoretical framework to understand and control edge states in non-Hermitian systems.

A novel transfer phenomenon was revealed, where edge modes with different sublattice symmetries localised at the same boundary, challenging the conventional understanding of edge state localisation and opening new possibilities for manipulating quantum information. The structure of the spectral loops in the complex quasienergy plane provided a clear signature for these transfer behaviours, offering a powerful tool for characterising and understanding the system’s behaviour. These findings deepen the understanding of non-unitary topology and offer valuable insights for the experimental realisation and control of edge states in non-Hermitian systems, paving the way for novel device designs and applications in wave manipulation and information processing. This research contributes to the growing body of work exploring the fascinating properties of non-Hermitian systems and their potential for technological innovation.